Let A = [ array cc 3 & -2 1 & 4 array ] , and let a = tr (A) and b = (A) . If is a complex root of the equation x^2 - x + 1 = 0 and m, n are real numbers such that m ^a + n ^b = 5 - 3 , then the value of 2m + 3n is equal to

Step-by-step solution

Given A = [ array cc 3 & -2 1 & 4 array ] . The trace of matrix A is a = 3 + 4 = 7 . The determinant of matrix A is b = (3)(4) - (-2)(1) = 12 + 2 = 14 . Since is a complex root of x^2 - x + 1 = 0 , we have ^2 - + 1 = 0 ^2 = - 1 . Multiplying by , we get ^3 = ^2 - = ( - 1) - = -1 . Now, consider the given equation: m ^a + n ^b = 5 - 3 Substituting a = 7 and b = 14 : m ^7 + n ¹⁴ = 5 - 3 We simplify the powers of using ^3 = -1 : ^7 = ( ^3)^2 = (-1)^2 = ¹⁴ = ( ^3)^4 ^2 = (-1)^4 ^2 = ^2 = - 1 Substituting these back int